Papers on dialogue categories

June 2006 - June 2012

A dialogue category is a symmetric monoidal category equipped with a notion of tensorial negation.
We establish that the free dialogue category is a category
of dialogue games and total innocent strategies.
The connection clarifies the algebraic and logical nature
of dialogue games, and their intrinsic connection to linear continuations.
The proof of the statement is based on an algebraic
presentation of dialogue categories inspired by knot theory,
and a difficult factorization theorem established by rewriting techniques.

June 2006 - June 2012

We introduce a two-sided version of dialogue categories
– called dialogue chiralities – formulated as an adjunction
between a monoidal category A and a monoidal category B
equivalent to the opposite category of A.
The two-sided formulation is compared to the original one-sided formulation
of dialogue categories by exhibiting a 2-dimensional equivalence
between a 2-category of dialogue categories and
a 2-category of dialogue chiralities.
The resulting coherence theorem clarifies
in what sense every dialogue chirality may be strictified to an equivalent
dialogue category.

June 2006 - September 2012

Every dialogue category comes equipped with a continuation monad
defined by applying the negation functor twice.
In this paper, we advocate that this double negation monad should be understood
as part of a larger parametric monad (or a lax action)
with parameter taken in the opposite of the dialogue category.
This alternative point of view has one main conceptual benefit:
it reveals that the strength of the continuation monad is
the fragment of a more fundamental and symmetric structure
--- provided by a distributivity law between the parametric continuation monad
and the canonical action of the dialogue category over itself.
The purpose of this work is to describe the formal properties
of this parametric continuation monad and of its distributivity law.

June 2006 - onwards

The purpose of this paper is to develop a combinatorial presentation of dialogue categories
based on the emergence of a fundamental symmetry of logic between proofs and anti-proofs. The micrological analysis reveals that tensorial negation
may be either taken as primitive, or decomposed into a series
of more elementary components described in the paper.

June 2006 - onwards

This paper is the second part of the micrological study of negation
initiated in our companion paper.
We have encountered there a discrepancy
between left and right dialogue chiralities
which we resolve here by introducing
the notion of ambidextrous dialogue chirality.
One main purpose of the paper is to disclose
the topological nature of this logical notion.
More specifically, we establish that an ambidextrous chirality
is the same thing as a left chirality equipped
with an helical structure on its tensorial negation.
This topological insight enables us to conclude the project
initiated in our companion paper, and to present
ambidextrous chiralities in a purely combinatorial way.

January 2009 - onwards

A dialogue category is a monoidal category equipped with an exponentiating object bot called its tensorial pole.
In a dialogue category, every object x is thus equipped with
a left negation x --o bot and a right negation bot o-- x.
An important point of the definition is that the object x is not required to coincide with its double negation.
Our main purpose in the present article is to formulate two non commutative notions
of dialogue categories -- called cyclic and balanced dialogue categories.
In particular, we show that the category of left H-modules
of arbitrary dimension on a ribbon Hopf algebra H defines
a balanced dialogue category Mod(H) whose
tensorial pole~$\bot$ is the underlying field k.
We explain how to recover from this basic observation the well-known fact that
the full subcategory of finite dimensional left H-modules defines a ribbon category.

7. A functorial bridge between proofs and knots

January 2009 - onwards

March 2012 - onwards

Starting from the particular case of the double negation monad
in dialogue categories, we investigate what additional structure
is required of an adjunction in order to give rise to a strong monad.
The analysis leads to a purely combinatorial description
of enriched functors, enriched natural transformations
and enriched adjunctions between categorical modules.

January 2009 - March 2013

About ten years ago, Brian Day and Ross Street discovered a beautiful and unexpected connection between the notion of star-autonomous category in proof theory and the notion of Frobenius algebra in mathematical physics. The purpose of the present paper is to clarify the logical content of this connection by formulating a two-sided presentation of Frobenius algebras. The presentation is inspired by the idea that every logical dispute has two sides consisting of a Prover and of a Denier. This dialogical point of view leads us to a correspondence between dialogue categories and Frobenius pseudomonoids. The correspondence with dialogue categories refines Day and Street’s correspondence with star-autonomous categories in the same way as tensorial logic refines linear logic.

10. Tensorial logic meets cobordism

January 2009 - onwards

May - September 2013

We explain how to see the set of positions of a dialogue game as a coherence space in the sense of Girard or as a bistructure in the sense of Curien, Plotkin and Winskel. The coherence structure on the set of positions results from a Kripke translation of tensorial logic into linear logic extended with a necessity modality. The translation is done in such a way that every innocent strategy defines a clique or a configuration in the resulting space of positions. This leads us to study the notion of configuration designed by Curien, Plotkin and Winskel for general bistructures in the particular case of a bistructure associated to a dialogue game. We show that every such configuration may be seen as an interactive strategy equipped with a backward as well as a forward dynamics based on the interplay between the stable order and the extensional order. In that way, the category of bistructures is shown to include a full subcategory of games and coherent strategies of an interesting nature.

Papers on resource modalities

The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of the misleading conception that linear logic is more primitive than game semantics. Here, we defend the opposite view, and thus advocate that game semantics is conceptually more primitive than linear logic. This revised point of view leads us to introduce tensor logic, a primitive variant of linear logic where negation is not involutive. After formulating its categorical semantics, we interpret tensor logic in a model based on Conway games equipped with a notion of payoff, in order to reflect the various resource policies of the logic: linear, affine, relevant or exponential.

The exponential modality of linear logic associates a commutative comonoid !A to every formula A, this enabling to duplicate the formula in the course of reasoning. Here, we explain how to compute the free commutative comonoid !A as a sequential limit of equalizers in any symmetric monoidal category where this sequential limit exists and commutes with the tensor product. We apply this general recipe to a series of models of linear logic, typically based on coherence spaces, Conway games and finiteness spaces. This algebraic description unifies for the first time the various constructions of the exponential modality in spaces and games. It also sheds light on the duplication policy of linear logic, and its interaction with classical duality and double negation completion.